[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 284

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MetaDriver >>:
Нету такой функции. Ну кроме y=0. Это моё заднее слово. :)

y=0 does not go into itself when rotated

 

Firstly, there is no 90-degree angle in the Olympiad problem. I did not know about the problem from "Quantum".

Secondly, judging by the sequence of questions, question a) is easier than the next one. So it is possible to prove something.

Thirdly, there is such a function - otherwise there would be no Olympiad problem :) It's just inertia of thinking that gets in the way.

Well, let's try to solve for 90 degrees, maybe some ideas will appear.

 
alsu >>:

y=0 не переходит в себя при повороте

Then there isn't any at all.

 

proof a)

It is easy to check that the point (a,b) always passes to the point (-b,a) when rotated by 90 degrees. Then when our function graph is rotated, its arbitrary point (x,f(x)) will change to (-f(x),x). But by the terms of the problem the new graph coincides with the old one, so we have to require

f(-f(x))=x (1)

for any x on the number axis. Now, if f(x0)=x0 is satisfied for some point x0, then according to (1) we should also satisfy f(-x0)=x0 (2)

Note that we can safely rotate the graph again by the same angle, and it will again pass into itself, but the point (-f(x),x) will already pass into (-x,-f(x)). So we have to assume that f(-x)=-f(x), with which (2) only agrees if x0=0, which was required to prove.


but I'm having a hard time with the example, too:))))


P.S. By the way, if you rotate it one more time, the proof is even more obvious, but that's lyric.

 
Mathemat >>:

Во-первых, в олимпиадной угла 90 градусов нет.

there are misprints too... the phrase "when turning an angle" looks suspicious, usually in the wording of problems if you want to indicate the uncertainty, use the phrase like "when turning a certain angle" or something like that... so I still vote for the typo.

 

So, point a) is solved for the special case. The fixed point is x=0.

OK, shall we look at the solution? I will only look at point a).

Yes, solution a) implicitly assumes that the angle is 90:

So, shall we keep the intrigue for point b)?

 

a) Crossing myself...:)


b) only making sure to cross yourself

 
alsu >>:

доказательство а)

нетрудно проверить, что точка (a,b) при повороте на 90 градусов всегда переходит в точку (-b,a). Тогда при повороте графика нашей функции произвольная его точка (x,f(x)) перейдет в (-f(x),x). Но по условию задачи новый график совпадает со старым, значит мы должны потребовать

f(-f(x))=x (1)

для любого x на числовой оси. Теперь, если для некой точки x0 выполняется f(x0)=x0, то согласно (1) должно выполняться и f(-x0)=x0 (2)

Заметим, что график мы можем спокойно вращать его еще раз на тот же угол, и он снова перейдет в себя, но при этом уже точка (-f(x),x) переходит в (-x,-f(x)). Значит мы обязаны принять, что f(-x)=-f(x), с чем (2) согласуется только в случае, если x0=0, что и требовалось доказать.


а вот с примером у меня тоже туговато:))))

I think I may have come up with an example. To be more exact, I have invented a way of constructing it. I will try to describe it (it is too complicated to draw, I was about to go to bed).

The function is, of course, discontinuous. So:

Draw a line y=x*1/2 (at an angle of Pi/6) through the origin. And another one: y=-x*2 (at an angle of -Pi/3).

These are the blanks. From them you need to cut pieces. We do it with a condition that at rotation the pieces coincide with their "doubles".

Next. Draw a vertical line to the right of the ordinate (for example x=1).

Take a compasses, put one leg to the origin, the second on the intersection point of the drawn vertical line with the first workpiece (x=1, y=0.5) and twist around O to intersect with the second workpiece. // However, it is better to rotate by all 360 - it will be useful in the future, for construction of the negative direction

(At x=0.5, y=-1)

From this intersection point, construct a vertical line up to the intersection with the first piece again (x=0.5, y=0.25)... and repeat the procedure once again. To the satisfaction, or rather infinitely.

The same is done in zoom direction (in reverse order, of course).

And now all the construction is duplicated in the negative direction.

That is all. The chart is ready. All that remains is to write the function it represents.

 
five points
 
alsu >>:
пять баллов

I'm like that myself! :)

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